Chaotic vibration of an autoparametrical system with the spherical pendulum

In the paper, the dynamics of a three degree of freedom vibratory system with a spherical
pendulum in the neighbourhood of internal and external resonance is considered. It
has been assumed that the spherical pendulum is suspended to the main body which is
then suspended to the element characterized by some elasticity and damping. The system
is excited harmonically in the vertical direction. The equation of motion has been solved
numerically. The influence of initial conditions on the behaviour of the spherical pendulum is
investigated. In this type of the system, one mode of vibration may excite or damp another
one, and for different kinds of periodic vibrations there may also appear chaotic vibrations.
For characterization of an irregular chaotic response, time histories, bifurcation diagrams,
power spectral densities, Poincar´e maps and the maximum Lyapunov exponents have been
calculated.